A mean-field description for the expansion kinetics of activated T cell populations.

When lymphocytes encounter their cognate antigen, they become activated and undergo a limited number of cell divisions during which they differentiate into memory or effector cells or die. While the dynamics of individual cells are often heterogeneous, the expansion kinetics at the population level are highly reproducible, suggesting a mean-field description. To generate a finite division destiny, we consider two scenarios: Cells stop dividing after a certain number of iterations or their death rate increases with each cell division. The dynamics of the combined system can be mapped to a partial differential equation, and for a suitable choice of the activation rate, we obtain simple analytical solutions for the total cell number and the mean number of divisions per cell which can well describe the signal-dependent T cell expansion kinetics from in vitro experiments. Interestingly, only the division cessation mechanism yields an expression for the division destiny that does not contradict experiments. We show that the generation-dependent decrease of the division rate in individual cells leads to a time-dependent decrease at the population level which is consistent with a "time-to-die" control mechanism for the division destiny as suggested previously. We also derive mean-field equations for the total cell number which provide a basis for implementing T cell expansion kinetics into quantitative systems pharmacology models for immuno-oncology and CAR-T cell therapies.

Virtual Populations for Quantitative Systems Pharmacology Models.

Quantitative systems pharmacology (QSP) places an emphasis on dynamic systems modeling, incorporating considerations from systems biology modeling and pharmacodynamics. The goal of QSP is often to quantitatively predict the effects of clinical therapeutics, their combinations, and their doses on clinical biomarkers and endpoints. In order to achieve this goal, strategies for incorporating clinical data into model calibration are critical. Virtual population (VPop) approaches facilitate model calibration while faced with challenges encountered in QSP model application, including modeling a breadth of clinical therapies, biomarkers, endpoints, utilizing data of varying structure and source, capturing observed clinical variability, and simulating with models that may require more substantial computational time and resources than often found in pharmacometrics applications. VPops are frequently developed in a process that may involve parameterization of isolated pathway models, integration into a larger QSP model, incorporation of clinical data, calibration, and quantitative validation that the model with the accompanying, calibrated VPop is suitable to address the intended question or help with the intended decision. Here, we introduce previous strategies for developing VPops in the context of a variety of therapeutic and safety areas: metabolic disorders, drug-induced liver injury, autoimmune diseases, and cancer. We introduce methodological considerations, prior work for sensitivity analysis and VPop algorithm design, and potential areas for future advancement. Finally, we give a more detailed application example of a VPop calibration algorithm that illustrates recent progress and many of the methodological considerations. In conclusion, although methodologies have varied, VPop strategies have been successfully applied to give valid clinical insights and predictions with the assistance of carefully defined and designed calibration and validation strategies. While a uniform VPop approach for all potential QSP applications may be challenging given the heterogeneity in use considerations, we anticipate continued innovation will help to drive VPop application for more challenging cases of greater scale while developing new rigorous methodologies and metrics.

Diffusive coupling can discriminate between similar reaction mechanisms in an allosteric enzyme system.

BACKGROUND: A central question for the understanding of biological reaction networks is how a particular dynamic behavior, such as bistability or oscillations, is realized at the molecular level. So far this question has been mainly addressed in well-mixed reaction systems which are conveniently described by ordinary differential equations. However, much less is known about how molecular details of a reaction mechanism can affect the dynamics in diffusively coupled systems because the resulting partial differential equations are much more difficult to analyze. RESULTS: Motivated by recent experiments we compare two closely related mechanisms for the product activation of allosteric enzymes with respect to their ability to induce different types of reaction-diffusion waves and stationary Turing patterns. The analysis is facilitated by mapping each model to an associated complex Ginzburg-Landau equation. We show that a sequential activation mechanism, as implemented in the model of Monod, Wyman and Changeux (MWC), can generate inward rotating spiral waves which were recently observed as glycolytic activity waves in yeast extracts. In contrast, in the limiting case of a simple Hill activation, the formation of inward propagating waves is suppressed by a Turing instability. The occurrence of this unusual wave dynamics is not related to the magnitude of the enzyme cooperativity (as it is true for the occurrence of oscillations), but to the sensitivity with respect to changes of the activator concentration. Also, the MWC mechanism generates wave patterns that are more stable against long wave length perturbations. CONCLUSIONS: This analysis demonstrates that amplitude equations, which describe the spatio-temporal dynamics near an instability, represent a valuable tool to investigate the molecular effects of reaction mechanisms on pattern formation in spatially extended systems. Using this approach we have shown that the occurrence of inward rotating spiral waves in glycolysis can be explained in terms of an MWC, but not with a Hill mechanism for the activation of the allosteric enzyme phosphofructokinase. Our results also highlight the importance of enzyme oligomerization for a possible experimental generation of Turing patterns in biological systems.

Modeling the electron transport chain of purple non-sulfur bacteria.

Purple non-sulfur bacteria (Rhodospirillaceae) have been extensively employed for studying principles of photosynthetic and respiratory electron transport phosphorylation and for investigating the regulation of gene expression in response to redox signals. Here, we use mathematical modeling to evaluate the steady-state behavior of the electron transport chain (ETC) in these bacteria under different environmental conditions. Elementary-modes analysis of a stoichiometric ETC model reveals nine operational modes. Most of them represent well-known functional states, however, two modes constitute reverse electron flow under respiratory conditions, which has been barely considered so far. We further present and analyze a kinetic model of the ETC in which rate laws of electron transfer steps are based on redox potential differences. Our model reproduces well-known phenomena of respiratory and photosynthetic operation of the ETC and also provides non-intuitive predictions. As one key result, model simulations demonstrate a stronger reduction of ubiquinone when switching from high-light to low-light conditions. This result is parameter insensitive and supports the hypothesis that the redox state of ubiquinone is a suitable signal for controlling photosynthetic gene expression.